A defaultable callable bond pricing model

Transcription

1 A deaultable callable bond pricing model AUTHOS ATILE INFO JOUNAL FOUNDE David Hua Heng-i ou David Wang David Hua, Heng-i ou and David Wang (009). A deaultable callable bond pricing model. Investment Management and Financial Innovations, 6(3) "Investment Management and Financial Innovations" LL onsulting Publising ompany Business Perspectives NUMBE OF EFEENES 0 NUMBE OF FIGUES 0 NUMBE OF TABLES 0 Te autor(s) 08. Tis publication is an open access article. businessperspectives.org

2 Investment Management and Financial Innovations, Volume 6, Issue 3, 009 David Hua (USA), Heng-i ou (Taiwan), David Wang (Taiwan) A deaultable callable bond pricing model Abstract Tis paper presents a 3D model or pricing deaultable bonds wit embedded call options. Te pricing model incorporates tree essential ingredients in te pricing o deaultable bonds: stocastic interest rate, stocastic deault ris, and call provision. Bot te stocastic interest rate and te stocastic deault ris are modeled as a square-root diusion process. Te deault ris process is allowed to be correlated wit te deault-ree term structure. Te call provision is modeled as a constraint on te value o te bond in te inite dierence sceme. Te numerical example sows tat te 3D model is capable o pricing deaultable bonds wit embedded call options adequately. Tis paper can provide new insigt or uture researc on deaultable bond pricing models. Keywords: deaultable bond, embedded option, square-root diusion process, partial dierential equation, inite dierence metod. JEL lassiication: 00, G3. Introduction Te pricing o deaultable securities as occupied a central place in te academic and practitioner literature. Te standard teoretical paradigm or pricing deaultable securities is te contingent claims approac pioneered by Blac and Scoles (973) []. Muc o te literature ollows Merton (974) [] by explicitly lining te ris o a irm s deault to te variability in te irm s asset value. Altoug tis line o researc as proven very useul in addressing te qualitatively important aspects o pricing deaultable securities, it as been less successul in practical applications. Te lac o success owes to te act tat irms capital structures are typically quite complex and priority rules are oten violated. In response to tese diiculties, an alternative modeling approac as been pursued in a number o papers, including Madan and Unal (994) [3], Jarrow and Turnbull (995) [4], Duie and Singleton (999) [5]. At eac instant, tere is some probability tat a irm deaults on its obligation. Tis is called te instantaneous probability o deault. Te processes o bot tis probability and te recovery rate determine te value o deault ris. Altoug tese processes are not ormally lined to te irm s asset value, tere is presumably some underlying relation, tus Duie and Singleton describe tis alternative approac as a reducedorm model (Duee, 999) [6]. Tis paper is an eort to develop one suc model in a 3D setting or pricing deaultable bonds wit embedded call options. Te remainder o tis paper is organized as ollows. Section presents te model. Section describes te metodology. Section 3 provides a numerical example. Te last section concludes tis paper. David Hua, Heng-i ou, David Wang, Model We derive te pricing model or deaultable bonds wit embedded call options by adopting Duie and Singleton (999) [5] s reduced-orm approac and Hull (000) [7] s replicating-portolio approac. According to Duie and Singleton, deaultable bonds can be valued by discounting at a deaultadusted interest rate, : r L, () were r is te ris-ree interest rate, is te azard rate or deault (i.e., te instantaneous probability o deault) at time t, and L is te loss rate (i.e., te expected ractional loss in te maret value) i deault was to occur at time t, conditional on te inormation available up to time t. Tat is, te price at time 0 o a deaultable discount bond,, is: T E[exp( dt) X ], () 0 were X is te ace value, T is te maturity time, and E is te ris-neutral, conditional expectation at date 0. Tis is natural, in tat L is te ris neutral meanloss rate o te deaultable discount bond due to deault. Discounting at te deault-adusted sortterm interest rate tereore accounts or bot te probability and timing o deault, as well as or te eect o losses on deault. A ey eature o Equation () is tat, assuming te ris neutral mean-loss rate process L being given exogenously, standard term-structure models or deault-ree debt are directly applicable to deaultable debt by parameterizing instead o r (Duie and Singleton, 999) [5]. We assume tat bot te deault-adusted interest rate and te azard rate it a ox, Ingersoll, and oss (I)-style model (985) [8], a square-root diusion model: d a ( b ) dt dz, (3)

3 d a ( b ) dt dz, (4) were dz and dz are Wiener processes, and te drit and te diusion parameters are constants and are assumed to be nown. Te I-style model incorporates mean reversion and ensures tat te deault-adusted interest rates and te azard rates are always non-negative. As or te loss rate L, it is assumed to be a constant. We mae te assumption tat tere are a total o tree deaultable bonds wose prices depend on te deault-adusted interest rate and te azard rate. Because te tree deaultable bonds are all dependent on te deault-adusted interest rate and te azard rate, it ollows rom Ito s lemma tat te price o te t deaultable bond,, ollows a diusion process: d were dt dz dz, (5) a t b a b, (6), (7). (8) In tese equations, is te instantaneous mean rate o return provided by, and are te components o te instantaneous standard deviation o te rate o return provided by tat may be attributed to and, and between dz and dz. is te correlation Because tere are tree deaultable bonds and two Wiener processes in equation (5), it is possible to orm an instantaneously risless portolio,, using te deaultable bonds. Deine as te amount o te t deaultable bond in te portolio, so tat. (9) Te must be cosen so tat te stocastic components o te returns rom te deaultable bonds are eliminated. From equation (5) tis means tat Investment Management and Financial Innovations, Volume 6, Issue 3, 009 0, (0) 0. () Te return rom te portolio is ten given by d dt. () Te cost o setting up te portolio is. I tere are no arbitrage opportunities, te portolio must earn te deault-adusted interest rate, so tat or (3) 0. (4) Equations (0), () and (4) can be regarded as tree omogeneous linear equations in te s. Te s are not all zero. From a well-nown teorem in linear algebra, equations (0), () and (4) can be consistent only i or (5) (6) or and tat are dependent only on te deaultadusted interest rate, te azard rate and time t. Substituting rom equations (6), (7) and (8) into equation (5), we obtain t a b a b tat reduces to (7) a b t a b 0. (8) 55

4 Investment Management and Financial Innovations, Volume 6, Issue 3, 009 Dropping te subscripts to, we deduce tat any deaultable bond wose price,, is contingent on te deault-adusted interest rate,, te azard rate,, and time, t, satisies te second-order dierential equation t 56 a b a b 0. Q.E.D. (9) On a coupon date, te bond value must ump by te amount o te coupon payment. Hence, to incorporate coupon payments into te model, we impose a ump condition:,,t,,t K, (0) were a coupon o K is received at time t. Bonds oten ave a call eature wic gives te issuing company te rigt to purcase bac te bond at any time during speciied periods or a speciied amount. According to te no-arbitrage argument, to incorporate a call eature into te model, we must impose a constraint on te bond s value:,,t D X D, () were X D is te call price and t D is te call date. To ind a unique solution o equation (9), we must impose one inal condition and our boundary conditions. Te inal condition corresponds to te payo at maturity and so or a coupon-paying bond:,,t P T KT, () were a principal amount o P T and a coupon payment o K T are received at maturity. Te irst boundary condition, wen te deaultadusted interest rate,, approaces zero percent, can be stated as: T t,,t,,t e,,t. (3) Te second boundary condition, wen te deaultadusted interest rate,, approaces ininity, can be stated as: T t,,t,,t e 0. (4) Te tird boundary condition, wen te azard rate,, approaces zero percent, can be stated as: T t,,t,,t e rlt t,,t e rt t,,t e. (5) Te ort boundary condition, wen te azard rate,, approaces ininity, can be stated as: T t,,t,,t e rlt t,,t e 0. (6). Metodology We solve te pricing model or deaultable bonds wit embedded call options by a 3D explicit inite dierence metod (Hull, 003 [9]; Wilmott, 000 [0]). Suppose tat te number o monts to maturity is T. We divide tis into L equally spaced intervals o lengt t = T / L. t is ixed at one mont. A total o L+ times are, tereore, considered: 0, t, t,, T. Suppose tat max is a azard rate suiciently ig tat, wen it is reaced, te bond as virtually no value. We deine = max / M and consider a total o M+ equally spaced azard rates: 0,,,, max. is set to be one percent. Suppose tat max is a deault-adusted interest rate suiciently ig tat, wen it is reaced, te bond as virtually no value. We deine = max / N and consider a total o N+ equally spaced deaultadusted interest rates: 0,,,, max. is set to be one percent. Te time points, azard rate points and deaultadusted interest rate points deine a 3D grid consisting o a total o (L+)(M+)(N+) points as sown in Figure. Te (, ) point on te 3D grid is te point tat corresponds to deault-adusted interest rate i, azard rate and time t. We use te variable, to denote te value o te bond at i te (, ) point. ecall tat te dierential equation or te price o a deaultable bond, (,, t), is given as:

5 a b t a b 0. (7) For an interior point (, ) in te 3D grid, can t be approximated by using a symmetric central dierence: = t t, (8) can be approximated by using a symmetric central dierence: i, = i,, (9) can be approximated by using a symmetric central dierence: =, (30) can be approximated by using a symmetric central dierence: =, (3) can be approximated by using a symmetric central dierence: = 4 Investment Management and Financial Innovations, Volume 6, Issue 3, 009, (3) and can be approximated by using a symmetric central dierence: =. (33) Substituting equations (8), (9), (30), (3), (3) and (33) into te dierential equation (7) and noting tat = i, = and = i,, te corresponding dierence equation can be sown as: t a b a b i i i i, i, i, i, i 4 i 0, (34) were i = 0,,, N, = 0,,, M and = 0,,, L. earranging terms, tis equation becomes: Bi Ai, D, (35) were A Bi i t t it, a b i i t i t a b t, t, D i t, 4 i = 0,,, N, = 0,,, M and = 0,,, L. Te value o te bond at time T is P T +K T, were P T is te principal amount and K T is te coupon payment. Hence,, P K (36) i T T or i = 0,,, N, = 0,,, M- and = 0. Te value o te bond wen te deault-adusted,,t. Hence, interest rate is zero percent is or i = 0, = 0,,, M- and = 0,,, L-. (37) 57

6 Investment Management and Financial Innovations, Volume 6, Issue 3, 009 We assume tat te bond is wort zero wen te deault-adusted interest rate is one undred percent, so tat 0 (38) 58 or i = N, = 0,,, M- and = 0,,, L-. Te value o te bond wen te azard rate is zero rt t percent is,,t e. Hence, e r T t or i =,,, N-, = 0 and = 0,,, L-. (39) We assume tat te bond is wort zero wen te azard rate is one undred percent, so tat 0 (40) or i = 0,,, N, = M and = -, 0,, L-. To incorporate coupon payments into te model, we impose a ump condition. Hence,, K (4) i i, or i = 0,,, N-, = 0,,, M-, = t or te coupon date and K is te coupon payment. To incorporate call eatures into te model, we impose a constraint on te bond s value. Hence,, X (4) i D or i = 0,,, N-, = 0,,, M-, = t D or te call date and X D is te call price. Equations (36), (37), (38), (39) and (40) deine te value o te bond along te ive planes o te 3D grid in Figure, were t = T, = 0%, = 00%, = 0% and = 00%. Equation (35) deines te value o te bond at all oter points. Equation (35) sows tat tere are nine nown bond values lined to one unnown bond value. See Figure. Hence, or eac time layer tere are (N-)(M- ) equations in (N-)(M-) unnowns; te boundary conditions yield te values at te our boundaries or eac time layer and te inal condition gives te values in te last time layer. To ind te bond value o interest, go bacwards in time, solving or a sequence o linear equations. L L L L Eventually,,,,,, 3,, N, M are obtained. One o tese is te bond price o interest. I te initial deault-adusted interest rate or te initial azard rate does not lie on te grid point, we use a linear interpolation between te two bond prices on te neigboring grid points to ind te bond price o interest. 3. Numerical example We validate te pricing model or deaultable bonds wit embedded call options by a numerical example. Te input data used or te model are summarized in Table. For te deault-adusted interest rate model, a = 0.35, b = 0.0, = 0.5 and = For te azard rate model, a = 0.30, b = 0.5, = 0.0 and = Te loss rate L is set to be Te correlation is set to be 0.0. Te bond to be priced is assumed to ave a maturity T o ten years. Te coupon payment K is set to be $0.00. Bot te principal amount P and te call price X are set to be $ We assume tat te coupon is paid semiannually in te 6t mont and te t mont eac year, and tat te bond is callable in te 3rd mont and te 9t mont o te 4t year, te 5t year, te 6t year and te 7t year. We irst compute te value o bot te straigt bond and te callable bond using dierent values o te ris-ree interest rate r. Intuitively, we expect tat as te value o r increases, te value o bot te straigt bond and te callable bond will decrease, and tat te value o te straigt bond will be greater tan te value o te callable bond. Te results are reported in Table and depicted in Figure 3. As expected, te results sow tat as te value o r increases, te value o bot te straigt bond and te callable bond decreases, and tat, or r less tan or equal to twentyive percent, te value o te straigt bond is greater tan te value o te callable bond. We also compute te value o te callable bond wit various numbers o call dates using dierent values o r. Wit one call date, te bond is callable in te 3rd mont o te 4t year; wit two call dates, te bond is callable in te 3rd mont and te 9t mont o te 4t year; wit tree call dates, te bond is callable in te 3rd mont and te 9t mont o te 4t year and te 3rd mont o te 5t year; wit our call dates, te bond is callable in te 3rd mont and te 9t mont o te 4t year and te 5t year; wit ive call dates, te bond is callable in te 3rd mont and te 9t mont o te 4t year and te 5t year and te 3rd mont o te 6t year. Intuitively, we expect tat as te number o call dates increases, te value o te callable bond will decrease. Te results are reported in Table 3 and depicted in Figure 4. As expected, te results sow tat, or r less tan or equal to ive percent, as te number o call dates increases, te value o te callable bond decreases. onclusion Tis paper presents a 3D model or pricing deaultable bonds wit embedded call options. Te

7 pricing model incorporates tree essential ingredients in te pricing o deaultable bonds: stocastic interest rate, stocastic deault ris, and call provision. Bot te stocastic interest rate and te stocastic deault ris are modeled as a square-root diusion process. Te deault ris process is allowed to be correlated wit te deault-ree term structure. Te call provision is modeled as a constraint on te value o te bond eerences Investment Management and Financial Innovations, Volume 6, Issue 3, 009 in te inite dierence sceme. Te numerical example sows tat te 3D model is capable o pricing deaultable bonds wit embedded call options adequately. Te model is by no means a complete success. To improve te model, one can assume tat te recovery rate in te event o deault varies stocastically troug time. In summary, tis paper can provide new insigt or uture researc on deaultable bond pricing models.. F. Blac and M. Scoles (973), Te pricing o options and corporate liabilities. Journal o Political Economy. 8: Merton (974), On te pricing o corporate debt: Te ris structure o interest rates. Journal o Finance. 9: D.B. Madan and H. Unal (994), Pricing te iss o Deault. Woring Paper, Warton Scool, University o Pennsylvania. 4..A. Jarrow and S.M. Turnbull (995), Pricing derivatives on inancial securities subect to credit ris. Journal o Finance. 50: D. Duie and K.J. Singleton (999), Modeling te term structure o deaultable bonds. eview o Financial Studies. : G. Duee (999), Estimating te price o deault ris. eview o Financial Studies. : J. Hull (000), Options, Futures, and Oter Derivatives. New Jersey: Prentice Hall. 8. J. ox, J. Ingersoll and S. oss (985), A teory o te term structure o interest rates. Econometrica. 53: J. Hull (003), Options, Futures, and Oter Derivatives. New Jersey: Prentice Hall. 0. P. Wilmott (000), Quantitative Finance. New Yor: Jon Wiley & Sons. Appendix Table. Te input data used or te model Deault-adusted interest rate model: Hazard rate model: everting speed a 0.35 everting speed a 0.30 everting level b 0.0 everting level b 0.5 Volatility 0.5 Volatility 0.0 Maret price o ris Maret price o ris Loss given deault: orrelation between and : Loss rate L 0.50 orrelation 0.0 Bond caracteristics: Maturity year T 0.00 Principal amount P $00.00 oupon payment K $0.00 all price X $00.00 Table. Te bond values obtained by te model or te straigt bond and te callable bond Interest rate Straigt bond allable bond 0% $ $ % $ $ % $ $ % $ $ % $ $ % $ $ % $ $ % $75.43 $

8 Investment Management and Financial Innovations, Volume 6, Issue 3, 009 Table (cont.). Te bond values obtained by te model or te straigt bond and te callable bond Interest rate Straigt bond allable bond 40% $ $ % $ $ % $ $ % $5.595 $ % $ $ % $58.46 $ % $ $ % $ $ % $5.535 $ % $ $ % $ $ % $ $ % $ $ Table 3. Te bond values obtained by te model or te callable bond wit various numbers o call dates Interest rate One call date Two call dates Tree call dates Four call dates Five call dates 0% $ $ $ $ $ % $ $ $.548 $.784 $ % $ $.3038 $.709 $ $ % $ $0.098 $ $7.393 $ % $ $ $.360 $ $ % $ $78.60 $ $ $ % $ $ $83.85 $ $ % $ $65.00 $68.04 $8.68 $ % $ $ $8.743 $ $ % $ $ $ $ $ % $ $65.75 $ $ $ % $ $ $ $ $ % $ $6.667 $ $ $ % $ $ $ $ $ % $ $3.5 $ $ $ % $ $ $8.64 $4.393 $ % $9.677 $ $ $9.87 $ % $3.678 $ $0.005 $3.03 $ % $9.35 $3.55 $3.346 $ $ % $ $ $3.543 $ $ % $ $ $ $ $

9 Investment Management and Financial Innovations, Volume 6, Issue 3, 009 Fig.. Te 3D inite dierence grid Fig.. Te relationsip between bond values in te 3D explicit inite dierence metod Fig. 3. Te bond values obtained by te model or te straigt bond and te callable bond 6

10 Investment Management and Financial Innovations, Volume 6, Issue 3, 009 Fig. 4. Te bond values obtained by te model or te callable bond wit various numbers o call dates 6